The richest class of t-perfect graphs known so far consists of the graphs with no so-called odd-K4. Clearly, these graphs have the special property that they are hereditary t-perfect in the sense that every subgraph is also t-perfect, but they are not the only ones. In this paper we characterize hereditary t-perfect graphs by showing that any non–t-perfect graph contains a non–tperfect subdivision of K4, called a bad-K4. To prove the result we show which “weakly 3-connected” graphs contain no bad-K4; as a side-product of this we get a polynomial time recognition algorithm. It should be noted that our result does not characterize t-perfection, as that is not maintained when taking subgraphs but only when taking induced subgraphs. AMS subject classifications. 05C75, 05C70, 90C10, 90C27 Key words. stable sets, polyhedra, odd circuits, decomposition PII. S0895480196306361
A. M. H. Gerards, F. Bruce Shepherd